Sum the series

Infinite sequences of numbers have always puzzled mathematicians, who have consensually developed the notion of convergence and limit. But everything changed when they became interested in the successive sums of these sequences, to which they gave the name of series. Convergent series (the "nice" ones), studied by all those destined for a scientific career, pose no problem. They allow defining the sum of an infinite number of terms, as long as the "partial sums" converge to a limit. But the others, officially "divergent" (the "wicked" ones), were at the origin of very different, sometimes astonishing approaches, creating true philosophical conflicts between famous scientists. Euler, Abel and some pioneers ventured to imagine defining sums for them, finding some wonders… which today are the subject of a beautiful theory!

What do Fermat's Last Theorem, Catalan's conjecture, counting rice grains on a chessboard, and Waring's problem have in common? They all involve powers, and they have occupied (and still occupy!) mathematicians for sometimes centuries. Numbers lend themselves well to this 'fifth arithmetic operation' that is raising to a given power. From then on, experiments flourished, from which some of the most famous conjectures and some unexpected applications would emerge. In cryptography, ensuring the confidentiality of exchanges requires using very large numbers, and exponentiation is a tool that makes it possible to obtain them with a moderate cost in terms of computation time.